Compound Interest

Percentage Accurate: 28.5% → 98.9%

Time: 12.2s

Alternatives: 12

Speedup: 8.1×

• could not determine a ground truth

  1. i = 1.743718544969785e-7
  2. n = 2.3365800898320574e+38

Specification

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 28.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))

C

double code(double i, double n) {
	return 100.0 * ((pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 100.0d0 * ((((1.0d0 + (i / n)) ** n) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	return 100.0 * ((Math.pow((1.0 + (i / n)), n) - 1.0) / (i / n));
}

Python

def code(i, n):
	return 100.0 * ((math.pow((1.0 + (i / n)), n) - 1.0) / (i / n))

Julia

function code(i, n)
	return Float64(100.0 * Float64(Float64((Float64(1.0 + Float64(i / n)) ^ n) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp = code(i, n)
	tmp = 100.0 * ((((1.0 + (i / n)) ^ n) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := N[(100.0 * N[(N[(N[Power[N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision], n], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := t\_0 - 1\\ t_2 := \frac{t\_1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot t\_1\right) \cdot n\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{i}, n, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (- t_0 1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 (- INFINITY))
     (* (* (/ 100.0 i) t_1) n)
     (if (<= t_2 0.0)
       (* (/ (* (expm1 (* (log1p (/ i n)) n)) 100.0) i) n)
       (if (<= t_2 INFINITY)
         (* (fma (/ t_0 i) n (/ (- n) i)) 100.0)
         (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = t_0 - 1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((100.0 / i) * t_1) * n;
	} else if (t_2 <= 0.0) {
		tmp = ((expm1((log1p((i / n)) * n)) * 100.0) / i) * n;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((t_0 / i), n, (-n / i)) * 100.0;
	} else {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(100.0 / i) * t_1) * n);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * 100.0) / i) * n);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(t_0 / i), n, Float64(Float64(-n) / i)) * 100.0);
	else
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(100.0 / i), $MachinePrecision] * t$95$1), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * 100.0), $MachinePrecision] / i), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n + N[((-n) / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -inf.0

   1. Initial program 100.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      10. lift--.f64 N/A

          \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \left(\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      13. lower-expm1.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot \frac{100}{i}\right) \cdot n \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right) \cdot \frac{100}{i}\right) \cdot n \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      16. lower-log1p.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      17. lower-/.f64 33.3

          \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

   4. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      3. lift-log1p.f64 N/A

         \[\leadsto \left(\left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      4. pow-to-exp N/A

         \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      6. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      8. +-commutative N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      10. lift-pow.f64 100.0

          \[\leadsto \left(\left(\color{blue}{{\left(\frac{i}{n} + 1\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

   6. Applied rewrites 100.0%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 24.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      10. lift--.f64 N/A

          \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \left(\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      13. lower-expm1.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot \frac{100}{i}\right) \cdot n \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right) \cdot \frac{100}{i}\right) \cdot n \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      16. lower-log1p.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      17. lower-/.f64 99.6

          \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right)} \cdot n \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i}} \cdot n \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i}} \cdot n \]

      5. lift-expm1.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1\right)} \cdot 100}{i} \cdot n \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{i} \cdot n \]

      7. lift-log1p.f64 N/A

         \[\leadsto \frac{\left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right) \cdot 100}{i} \cdot n \]

      8. pow-to-exp N/A

         \[\leadsto \frac{\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot 100}{i} \cdot n \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot 100}{i} \cdot n \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot 100}{i} \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \frac{\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{i} \cdot n \]

      13. lift-log1p.f64 N/A

          \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n} - 1\right) \cdot 100}{i} \cdot n \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right) \cdot 100}{i} \cdot n \]

      15. lift-expm1.f64 99.7

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right)} \cdot 100}{i} \cdot n \]

   6. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i}} \cdot n \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      16. lower-neg.f64 99.9

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 81.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot 100}{i} \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{i}{n} + 1\right)}^{n}\\ t_1 := t\_0 - 1\\ t_2 := \frac{t\_1}{\frac{i}{n}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot t\_1\right) \cdot n\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{i}, n, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (pow (+ (/ i n) 1.0) n)) (t_1 (- t_0 1.0)) (t_2 (/ t_1 (/ i n))))
   (if (<= t_2 (- INFINITY))
     (* (* (/ 100.0 i) t_1) n)
     (if (<= t_2 0.0)
       (* (* (expm1 (* (log1p (/ i n)) n)) (/ 100.0 i)) n)
       (if (<= t_2 INFINITY)
         (* (fma (/ t_0 i) n (/ (- n) i)) 100.0)
         (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n))))))

C

double code(double i, double n) {
	double t_0 = pow(((i / n) + 1.0), n);
	double t_1 = t_0 - 1.0;
	double t_2 = t_1 / (i / n);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((100.0 / i) * t_1) * n;
	} else if (t_2 <= 0.0) {
		tmp = (expm1((log1p((i / n)) * n)) * (100.0 / i)) * n;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((t_0 / i), n, (-n / i)) * 100.0;
	} else {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(i / n) + 1.0) ^ n
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(t_1 / Float64(i / n))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(100.0 / i) * t_1) * n);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(expm1(Float64(log1p(Float64(i / n)) * n)) * Float64(100.0 / i)) * n);
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(t_0 / i), n, Float64(Float64(-n) / i)) * 100.0);
	else
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[Power[N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision], n], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(i / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(100.0 / i), $MachinePrecision] * t$95$1), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(Exp[N[(N[Log[1 + N[(i / n), $MachinePrecision]], $MachinePrecision] * n), $MachinePrecision]] - 1), $MachinePrecision] * N[(100.0 / i), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(t$95$0 / i), $MachinePrecision] * n + N[((-n) / i), $MachinePrecision]), $MachinePrecision] * 100.0), $MachinePrecision], N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < -inf.0

   1. Initial program 100.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      10. lift--.f64 N/A

          \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \left(\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      13. lower-expm1.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot \frac{100}{i}\right) \cdot n \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right) \cdot \frac{100}{i}\right) \cdot n \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      16. lower-log1p.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      17. lower-/.f64 33.3

          \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

   4. Applied rewrites 33.3%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(e^{\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      3. lift-log1p.f64 N/A

         \[\leadsto \left(\left(e^{\color{blue}{\log \left(1 + \frac{i}{n}\right)} \cdot n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      4. pow-to-exp N/A

         \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      6. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left({\left(1 + \color{blue}{\frac{i}{n}}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      8. +-commutative N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left({\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      10. lift-pow.f64 100.0

          \[\leadsto \left(\left(\color{blue}{{\left(\frac{i}{n} + 1\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

   6. Applied rewrites 100.0%

      \[\leadsto \left(\color{blue}{\left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

   if -inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < 0.0

   1. Initial program 24.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\frac{i}{n}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{\color{blue}{\frac{i}{n}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)}{i} \cdot n} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot 100}}{i} \cdot n \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right) \cdot \frac{100}{i}\right)} \cdot n \]

      10. lift--.f64 N/A

          \[\leadsto \left(\color{blue}{\left({\left(1 + \frac{i}{n}\right)}^{n} - 1\right)} \cdot \frac{100}{i}\right) \cdot n \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\left(\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      12. pow-to-exp N/A

          \[\leadsto \left(\left(\color{blue}{e^{\log \left(1 + \frac{i}{n}\right) \cdot n}} - 1\right) \cdot \frac{100}{i}\right) \cdot n \]

      13. lower-expm1.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{i}{n}\right) \cdot n\right)} \cdot \frac{100}{i}\right) \cdot n \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{i}{n}\right) \cdot n}\right) \cdot \frac{100}{i}\right) \cdot n \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\log \color{blue}{\left(1 + \frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      16. lower-log1p.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{i}{n}\right)} \cdot n\right) \cdot \frac{100}{i}\right) \cdot n \]

      17. lower-/.f64 99.6

          \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \color{blue}{\frac{100}{i}}\right) \cdot n \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n} \]

   if 0.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n)) < +inf.0

   1. Initial program 99.8%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i}}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(1 + \frac{i}{n}\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{i}{n} + 1\right)}}^{n}}{i}, n, \mathsf{neg}\left(\frac{n}{i}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \color{blue}{\frac{\mathsf{neg}\left(n\right)}{i}}\right) \]

      16. lower-neg.f64 99.9

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{\color{blue}{-n}}{i}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right)} \]

   if +inf.0 < (/.f64 (-.f64 (pow.f64 (+.f64 #s(literal 1 binary64) (/.f64 i n)) n) #s(literal 1 binary64)) (/.f64 i n))

   1. Initial program 0.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 81.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq -\infty:\\ \;\;\;\;\left(\frac{100}{i} \cdot \left({\left(\frac{i}{n} + 1\right)}^{n} - 1\right)\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq 0:\\ \;\;\;\;\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{n}\right) \cdot n\right) \cdot \frac{100}{i}\right) \cdot n\\ \mathbf{elif}\;\frac{{\left(\frac{i}{n} + 1\right)}^{n} - 1}{\frac{i}{n}} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i}, n, \frac{-n}{i}\right) \cdot 100\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{if}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (* (/ (expm1 i) i) 100.0) n)))
   (if (<= n -3e-240)
     t_0
     (if (<= n 4.7e-220)
       (* (/ (* n n) n) 100.0)
       (if (<= n 9.2e-17)
         (*
          (/
           1.0
           (fma
            (fma
             (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
             i
             -0.005)
            i
            0.01))
          n)
         t_0)))))

C

double code(double i, double n) {
	double t_0 = ((expm1(i) / i) * 100.0) * n;
	double tmp;
	if (n <= -3e-240) {
		tmp = t_0;
	} else if (n <= 4.7e-220) {
		tmp = ((n * n) / n) * 100.0;
	} else if (n <= 9.2e-17) {
		tmp = (1.0 / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(Float64(expm1(i) / i) * 100.0) * n)
	tmp = 0.0
	if (n <= -3e-240)
		tmp = t_0;
	elseif (n <= 4.7e-220)
		tmp = Float64(Float64(Float64(n * n) / n) * 100.0);
	elseif (n <= 9.2e-17)
		tmp = Float64(Float64(1.0 / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(N[(Exp[i] - 1), $MachinePrecision] / i), $MachinePrecision] * 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -3e-240], t$95$0, If[LessEqual[n, 4.7e-220], N[(N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 9.2e-17], N[(N[(1.0 / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.99999999999999991e-240 or 9.20000000000000035e-17 < n

   1. Initial program 25.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 89.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   if -2.99999999999999991e-240 < n < 4.7000000000000003e-220

   1. Initial program 68.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n\right)} \]

   5. Applied rewrites 0.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, i, -0.5\right), i, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(n + i \cdot \left(\frac{1}{2} \cdot n - \frac{1}{2}\right), n, \left(i \cdot i\right) \cdot \frac{1}{3}\right)}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.6%

       \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n, -0.5\right), i, n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{n} \]

   12. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   if 4.7000000000000003e-220 < n < 9.20000000000000035e-17

   1. Initial program 14.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 54.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.0%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 78.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{if}\;n \leq -2.8 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0
         (*
          (/
           1.0
           (fma
            (fma
             (fma (* i i) -1.388888888888889e-5 0.0008333333333333334)
             i
             -0.005)
            i
            0.01))
          n)))
   (if (<= n -2.8e+120)
     (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
     (if (<= n -3e-240)
       t_0
       (if (<= n 4.7e-220)
         (* (/ (* n n) n) 100.0)
         (if (<= n 9.2e-17)
           t_0
           (*
            (fma
             (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
             i
             1.0)
            (* 100.0 n))))))))

C

double code(double i, double n) {
	double t_0 = (1.0 / fma(fma(fma((i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n;
	double tmp;
	if (n <= -2.8e+120) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -3e-240) {
		tmp = t_0;
	} else if (n <= 4.7e-220) {
		tmp = ((n * n) / n) * 100.0;
	} else if (n <= 9.2e-17) {
		tmp = t_0;
	} else {
		tmp = fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(Float64(1.0 / fma(fma(fma(Float64(i * i), -1.388888888888889e-5, 0.0008333333333333334), i, -0.005), i, 0.01)) * n)
	tmp = 0.0
	if (n <= -2.8e+120)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -3e-240)
		tmp = t_0;
	elseif (n <= 4.7e-220)
		tmp = Float64(Float64(Float64(n * n) / n) * 100.0);
	elseif (n <= 9.2e-17)
		tmp = t_0;
	else
		tmp = Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(N[(N[(i * i), $MachinePrecision] * -1.388888888888889e-5 + 0.0008333333333333334), $MachinePrecision] * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -2.8e+120], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -3e-240], t$95$0, If[LessEqual[n, 4.7e-220], N[(N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 9.2e-17], t$95$0, N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.8000000000000001e120

   1. Initial program 8.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 97.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 97.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 74.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -2.8000000000000001e120 < n < -2.99999999999999991e-240 or 4.7000000000000003e-220 < n < 9.20000000000000035e-17

   1. Initial program 30.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 71.0

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.9%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(i \cdot \left(\frac{1}{1200} + \frac{-1}{72000} \cdot {i}^{2}\right) - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 74.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n \]

   if -2.99999999999999991e-240 < n < 4.7000000000000003e-220

   1. Initial program 68.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n\right)} \]

   5. Applied rewrites 0.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, i, -0.5\right), i, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(n + i \cdot \left(\frac{1}{2} \cdot n - \frac{1}{2}\right), n, \left(i \cdot i\right) \cdot \frac{1}{3}\right)}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.6%

       \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n, -0.5\right), i, n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{n} \]

   12. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   if 9.20000000000000035e-17 < n

   1. Initial program 25.4%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 93.9

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot 100\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(i \cdot i, -1.388888888888889 \cdot 10^{-5}, 0.0008333333333333334\right), i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.6e+188)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -3e-240)
     (* (/ 1.0 (fma -0.005 i 0.01)) n)
     (if (<= n 4.7e-220)
       (* (/ (* n n) n) 100.0)
       (if (<= n 7.2e-13)
         (* (/ 1.0 (fma (fma 0.0008333333333333334 i -0.005) i 0.01)) n)
         (*
          (fma
           (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
           i
           1.0)
          (* 100.0 n)))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.6e+188) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -3e-240) {
		tmp = (1.0 / fma(-0.005, i, 0.01)) * n;
	} else if (n <= 4.7e-220) {
		tmp = ((n * n) / n) * 100.0;
	} else if (n <= 7.2e-13) {
		tmp = (1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n;
	} else {
		tmp = fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.6e+188)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -3e-240)
		tmp = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n);
	elseif (n <= 4.7e-220)
		tmp = Float64(Float64(Float64(n * n) / n) * 100.0);
	elseif (n <= 7.2e-13)
		tmp = Float64(Float64(1.0 / fma(fma(0.0008333333333333334, i, -0.005), i, 0.01)) * n);
	else
		tmp = Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.6e+188], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -3e-240], N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 4.7e-220], N[(N[(N[(n * n), $MachinePrecision] / n), $MachinePrecision] * 100.0), $MachinePrecision], If[LessEqual[n, 7.2e-13], N[(N[(1.0 / N[(N[(0.0008333333333333334 * i + -0.005), $MachinePrecision] * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if n < -5.5999999999999996e188

   1. Initial program 10.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 96.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -5.5999999999999996e188 < n < -2.99999999999999991e-240

   1. Initial program 31.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 70.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -2.99999999999999991e-240 < n < 4.7000000000000003e-220

   1. Initial program 68.9%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \color{blue}{\left(n + i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(i \cdot \left(i \cdot \left(n \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{n}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{n}\right)\right) + n\right)} \]

   5. Applied rewrites 0.7%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(i \cdot n, \mathsf{fma}\left(\left(\frac{0.3333333333333333}{n \cdot n} + 0.16666666666666666\right) - \frac{0.5}{n}, i, 0.5 - \frac{0.5}{n}\right), n\right)} \]

   6. Taylor expanded in n around 0

      \[\leadsto 100 \cdot \frac{\frac{1}{3} \cdot {i}^{2} + n \cdot \left(i \cdot \left(\frac{-1}{2} \cdot i - \frac{1}{2}\right) + n \cdot \left(1 + i \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot i\right)\right)\right)}{\color{blue}{n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.6%

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, i, -0.5\right), i, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, i, 0.5\right), i, 1\right) \cdot n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{\color{blue}{n}} \]

   9. Taylor expanded in i around 0

      \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(n + i \cdot \left(\frac{1}{2} \cdot n - \frac{1}{2}\right), n, \left(i \cdot i\right) \cdot \frac{1}{3}\right)}{n} \]
   10. Step-by-step derivation

   11. Applied rewrites 24.6%

       \[\leadsto 100 \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, n, -0.5\right), i, n\right), n, \left(i \cdot i\right) \cdot 0.3333333333333333\right)}{n} \]

   12. Taylor expanded in i around 0

       \[\leadsto 100 \cdot \frac{{n}^{2}}{n} \]
   13. Step-by-step derivation

   14. Applied rewrites 82.6%

       \[\leadsto 100 \cdot \frac{n \cdot n}{n} \]

   if 4.7000000000000003e-220 < n < 7.1999999999999996e-13

   1. Initial program 13.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 57.3

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 57.3%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.2%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + i \cdot \left(\frac{1}{1200} \cdot i - \frac{1}{200}\right)} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 76.5%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n \]

   if 7.1999999999999996e-13 < n

   1. Initial program 26.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 94.3

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot 100\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 83.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 83.2%

       \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 4.7 \cdot 10^{-220}:\\ \;\;\;\;\frac{n \cdot n}{n} \cdot 100\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.0008333333333333334, i, -0.005\right), i, 0.01\right)} \cdot n\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.6e+188)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -3e-240)
     (* (/ 1.0 (fma -0.005 i 0.01)) n)
     (if (<= n 2.15e-128)
       0.0
       (*
        (fma
         (fma (fma 0.041666666666666664 i 0.16666666666666666) i 0.5)
         i
         1.0)
        (* 100.0 n))))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.6e+188) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -3e-240) {
		tmp = (1.0 / fma(-0.005, i, 0.01)) * n;
	} else if (n <= 2.15e-128) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * (100.0 * n);
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.6e+188)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -3e-240)
		tmp = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n);
	elseif (n <= 2.15e-128)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(fma(0.041666666666666664, i, 0.16666666666666666), i, 0.5), i, 1.0) * Float64(100.0 * n));
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.6e+188], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -3e-240], N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.15e-128], 0.0, N[(N[(N[(N[(0.041666666666666664 * i + 0.16666666666666666), $MachinePrecision] * i + 0.5), $MachinePrecision] * i + 1.0), $MachinePrecision] * N[(100.0 * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.5999999999999996e188

   1. Initial program 10.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 96.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -5.5999999999999996e188 < n < -2.99999999999999991e-240

   1. Initial program 31.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 70.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -2.99999999999999991e-240 < n < 2.14999999999999997e-128

   1. Initial program 46.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 10.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{0} \]

   if 2.14999999999999997e-128 < n

   1. Initial program 22.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 88.7

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(\left(1 + i \cdot \left(\frac{1}{2} + i \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot i\right)\right)\right) \cdot 100\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot 100\right) \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 80.2%

       \[\leadsto \color{blue}{\left(n \cdot 100\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, i, 0.16666666666666666\right), i, 0.5\right), i, 1\right) \cdot \left(100 \cdot n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= n -5.6e+188)
   (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)
   (if (<= n -3e-240)
     (* (/ 1.0 (fma -0.005 i 0.01)) n)
     (if (<= n 2.15e-128)
       0.0
       (*
        (fma (fma (fma 4.166666666666667 i 16.666666666666668) i 50.0) i 100.0)
        n)))))

C

double code(double i, double n) {
	double tmp;
	if (n <= -5.6e+188) {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	} else if (n <= -3e-240) {
		tmp = (1.0 / fma(-0.005, i, 0.01)) * n;
	} else if (n <= 2.15e-128) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (n <= -5.6e+188)
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	elseif (n <= -3e-240)
		tmp = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n);
	elseif (n <= 2.15e-128)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(fma(4.166666666666667, i, 16.666666666666668), i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[n, -5.6e+188], N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, -3e-240], N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.15e-128], 0.0, N[(N[(N[(N[(4.166666666666667 * i + 16.666666666666668), $MachinePrecision] * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if n < -5.5999999999999996e188

   1. Initial program 10.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 96.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -5.5999999999999996e188 < n < -2.99999999999999991e-240

   1. Initial program 31.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 70.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -2.99999999999999991e-240 < n < 2.14999999999999997e-128

   1. Initial program 46.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 10.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{0} \]

   if 2.14999999999999997e-128 < n

   1. Initial program 22.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 88.7

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + i \cdot \left(\frac{50}{3} + \frac{25}{6} \cdot i\right)\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.166666666666667, i, 16.666666666666668\right), i, 50\right), i, 100\right) \cdot n \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \mathbf{if}\;n \leq -5.6 \cdot 10^{+188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -3 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))
   (if (<= n -5.6e+188)
     t_0
     (if (<= n -3e-240)
       (* (/ 1.0 (fma -0.005 i 0.01)) n)
       (if (<= n 2.15e-128) 0.0 t_0)))))

C

double code(double i, double n) {
	double t_0 = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	double tmp;
	if (n <= -5.6e+188) {
		tmp = t_0;
	} else if (n <= -3e-240) {
		tmp = (1.0 / fma(-0.005, i, 0.01)) * n;
	} else if (n <= 2.15e-128) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(i, n)
	t_0 = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n)
	tmp = 0.0
	if (n <= -5.6e+188)
		tmp = t_0;
	elseif (n <= -3e-240)
		tmp = Float64(Float64(1.0 / fma(-0.005, i, 0.01)) * n);
	elseif (n <= 2.15e-128)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]}, If[LessEqual[n, -5.6e+188], t$95$0, If[LessEqual[n, -3e-240], N[(N[(1.0 / N[(-0.005 * i + 0.01), $MachinePrecision]), $MachinePrecision] * n), $MachinePrecision], If[LessEqual[n, 2.15e-128], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.5999999999999996e188 or 2.14999999999999997e-128 < n

   1. Initial program 19.1%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 90.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 75.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]

   if -5.5999999999999996e188 < n < -2.99999999999999991e-240

   1. Initial program 31.6%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 84.1

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.1%

      \[\leadsto \frac{1}{\frac{i}{\mathsf{expm1}\left(i\right) \cdot 100}} \cdot n \]

   8. Taylor expanded in i around 0

      \[\leadsto \frac{1}{\frac{1}{100} + \frac{-1}{200} \cdot i} \cdot n \]
   9. Step-by-step derivation

   10. Applied rewrites 70.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-0.005, i, 0.01\right)} \cdot n \]

   if -2.99999999999999991e-240 < n < 2.14999999999999997e-128

   1. Initial program 46.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 10.8%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 66.7

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 66.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.2e+58) 0.0 (* (fma (fma 16.666666666666668 i 50.0) i 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.2e+58) {
		tmp = 0.0;
	} else {
		tmp = fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.2e+58)
		tmp = 0.0;
	else
		tmp = Float64(fma(fma(16.666666666666668, i, 50.0), i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.2e+58], 0.0, N[(N[(N[(16.666666666666668 * i + 50.0), $MachinePrecision] * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -4.20000000000000024e58

   1. Initial program 73.7%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 67.2%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 39.8

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.8%

      \[\leadsto \color{blue}{0} \]

   if -4.20000000000000024e58 < i

   1. Initial program 20.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 79.8

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + i \cdot \left(50 + \frac{50}{3} \cdot i\right)\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 72.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(16.666666666666668, i, 50\right), i, 100\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;0\\ \mathbf{elif}\;i \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;100 \cdot n\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -4.2e+58) 0.0 (if (<= i 1.6e+21) (* 100.0 n) 0.0)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -4.2e+58) {
		tmp = 0.0;
	} else if (i <= 1.6e+21) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: tmp
    if (i <= (-4.2d+58)) then
        tmp = 0.0d0
    else if (i <= 1.6d+21) then
        tmp = 100.0d0 * n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double i, double n) {
	double tmp;
	if (i <= -4.2e+58) {
		tmp = 0.0;
	} else if (i <= 1.6e+21) {
		tmp = 100.0 * n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(i, n):
	tmp = 0
	if i <= -4.2e+58:
		tmp = 0.0
	elif i <= 1.6e+21:
		tmp = 100.0 * n
	else:
		tmp = 0.0
	return tmp

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -4.2e+58)
		tmp = 0.0;
	elseif (i <= 1.6e+21)
		tmp = Float64(100.0 * n);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(i, n)
	tmp = 0.0;
	if (i <= -4.2e+58)
		tmp = 0.0;
	elseif (i <= 1.6e+21)
		tmp = 100.0 * n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[i_, n_] := If[LessEqual[i, -4.2e+58], 0.0, If[LessEqual[i, 1.6e+21], N[(100.0 * n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if i < -4.20000000000000024e58 or 1.6e21 < i

   1. Initial program 60.2%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 53.6%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 32.0

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 32.0%

      \[\leadsto \color{blue}{0} \]

   if -4.20000000000000024e58 < i < 1.6e21

   1. Initial program 10.5%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot n} \]
   4. Step-by-step derivation

      1. lower-*.f64 80.2

         \[\leadsto \color{blue}{100 \cdot n} \]

   5. Applied rewrites 80.2%

      \[\leadsto \color{blue}{100 \cdot n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -1.55:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(50, i, 100\right) \cdot n\\ \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (if (<= i -1.55) 0.0 (* (fma 50.0 i 100.0) n)))

C

double code(double i, double n) {
	double tmp;
	if (i <= -1.55) {
		tmp = 0.0;
	} else {
		tmp = fma(50.0, i, 100.0) * n;
	}
	return tmp;
}

Julia

function code(i, n)
	tmp = 0.0
	if (i <= -1.55)
		tmp = 0.0;
	else
		tmp = Float64(fma(50.0, i, 100.0) * n);
	end
	return tmp
end

Wolfram

code[i_, n_] := If[LessEqual[i, -1.55], 0.0, N[(N[(50.0 * i + 100.0), $MachinePrecision] * n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -1.55000000000000004

   1. Initial program 62.0%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

      2. lift--.f64 N/A

         \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

      3. div-sub N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

      5. clear-num N/A

         \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

      6. sub-neg N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      8. clear-num N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      9. associate-/r/ N/A

         \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      11. distribute-frac-neg2 N/A

          \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

      13. frac-2neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      16. metadata-eval N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

      18. associate-/r/ N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   4. Applied rewrites 57.0%

      \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

   5. Taylor expanded in i around 0

      \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

      3. metadata-eval N/A

         \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

      4. mul0-lft N/A

         \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{i} \]

      6. lower-/.f64 32.6

         \[\leadsto \color{blue}{\frac{0}{i}} \]

   7. Applied rewrites 32.6%

      \[\leadsto \color{blue}{\frac{0}{i}} \]
   8. Step-by-step derivation

   9. Applied rewrites 32.6%

      \[\leadsto \color{blue}{0} \]

   if -1.55000000000000004 < i

   1. Initial program 20.3%

      \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{100 \cdot \frac{n \cdot \left(e^{i} - 1\right)}{i}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto 100 \cdot \color{blue}{\left(n \cdot \frac{e^{i} - 1}{i}\right)} \]

      2. *-commutative N/A

         \[\leadsto 100 \cdot \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot n\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(100 \cdot \frac{e^{i} - 1}{i}\right) \cdot n} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{e^{i} - 1}{i} \cdot 100\right)} \cdot n \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{e^{i} - 1}{i}} \cdot 100\right) \cdot n \]

      8. lower-expm1.f64 79.4

         \[\leadsto \left(\frac{\color{blue}{\mathsf{expm1}\left(i\right)}}{i} \cdot 100\right) \cdot n \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(i\right)}{i} \cdot 100\right) \cdot n} \]

   6. Taylor expanded in i around 0

      \[\leadsto \left(100 + 50 \cdot i\right) \cdot n \]
   7. Step-by-step derivation

   8. Applied rewrites 73.0%

      \[\leadsto \mathsf{fma}\left(50, i, 100\right) \cdot n \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 18.2% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (i n) :precision binary64 0.0)

C

double code(double i, double n) {
	return 0.0;
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double i, double n) {
	return 0.0;
}

Python

def code(i, n):
	return 0.0

Julia

function code(i, n)
	return 0.0
end

MATLAB

function tmp = code(i, n)
	tmp = 0.0;
end

Wolfram

code[i_, n_] := 0.0

Derivation

1. Initial program 27.8%

   \[100 \cdot \frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto 100 \cdot \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n} - 1}{\frac{i}{n}}} \]

   2. lift--.f64 N/A

      \[\leadsto 100 \cdot \frac{\color{blue}{{\left(1 + \frac{i}{n}\right)}^{n} - 1}}{\frac{i}{n}} \]

   3. div-sub N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\frac{i}{n}}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \frac{1}{\color{blue}{\frac{i}{n}}}\right) \]

   5. clear-num N/A

      \[\leadsto 100 \cdot \left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} - \color{blue}{\frac{n}{i}}\right) \]

   6. sub-neg N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}} + \left(\mathsf{neg}\left(\frac{n}{i}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto 100 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{n}{i}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

   8. clear-num N/A

      \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{i}{n}}}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   9. associate-/r/ N/A

      \[\leadsto 100 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{i} \cdot n}\right)\right) + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto 100 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{i}\right)\right) \cdot n} + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   11. distribute-frac-neg2 N/A

       \[\leadsto 100 \cdot \left(\color{blue}{\frac{1}{\mathsf{neg}\left(i\right)}} \cdot n + \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(i\right)}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right)} \]

   13. frac-2neg N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(i\right)\right)\right)}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   14. remove-double-neg N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   15. lower-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{i}}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   16. metadata-eval N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{\color{blue}{-1}}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\frac{i}{n}}\right) \]

   17. lift-/.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(1 + \frac{i}{n}\right)}^{n}}{\color{blue}{\frac{i}{n}}}\right) \]

   18. associate-/r/ N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto 100 \cdot \mathsf{fma}\left(\frac{-1}{i}, n, \color{blue}{\frac{{\left(1 + \frac{i}{n}\right)}^{n}}{i} \cdot n}\right) \]

4. Applied rewrites 21.7%

   \[\leadsto 100 \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{i}, n, \frac{{\left(\frac{i}{n} + 1\right)}^{n}}{i} \cdot n\right)} \]

5. Taylor expanded in i around 0

   \[\leadsto \color{blue}{100 \cdot \frac{n + -1 \cdot n}{i}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{100 \cdot \left(n + -1 \cdot n\right)}{i}} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \frac{100 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot n\right)}}{i} \]

   3. metadata-eval N/A

      \[\leadsto \frac{100 \cdot \left(\color{blue}{0} \cdot n\right)}{i} \]

   4. mul0-lft N/A

      \[\leadsto \frac{100 \cdot \color{blue}{0}}{i} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{i} \]

   6. lower-/.f64 17.9

      \[\leadsto \color{blue}{\frac{0}{i}} \]

7. Applied rewrites 17.9%

   \[\leadsto \color{blue}{\frac{0}{i}} \]
8. Step-by-step derivation

9. Applied rewrites 17.9%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 35.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{i}{n}\\ 100 \cdot \frac{e^{n \cdot \begin{array}{l} \mathbf{if}\;t\_0 = 1:\\ \;\;\;\;\frac{i}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{n} \cdot \log t\_0}{\left(\frac{i}{n} + 1\right) - 1}\\ \end{array}} - 1}{\frac{i}{n}} \end{array} \end{array} \]

FPCore

(FPCore (i n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ i n))))
   (*
    100.0
    (/
     (-
      (exp
       (*
        n
        (if (== t_0 1.0)
          (/ i n)
          (/ (* (/ i n) (log t_0)) (- (+ (/ i n) 1.0) 1.0)))))
      1.0)
     (/ i n)))))

C

double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
}

Fortran

real(8) function code(i, n)
    real(8), intent (in) :: i
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (i / n)
    if (t_0 == 1.0d0) then
        tmp = i / n
    else
        tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0d0) - 1.0d0)
    end if
    code = 100.0d0 * ((exp((n * tmp)) - 1.0d0) / (i / n))
end function

Java

public static double code(double i, double n) {
	double t_0 = 1.0 + (i / n);
	double tmp;
	if (t_0 == 1.0) {
		tmp = i / n;
	} else {
		tmp = ((i / n) * Math.log(t_0)) / (((i / n) + 1.0) - 1.0);
	}
	return 100.0 * ((Math.exp((n * tmp)) - 1.0) / (i / n));
}

Python

def code(i, n):
	t_0 = 1.0 + (i / n)
	tmp = 0
	if t_0 == 1.0:
		tmp = i / n
	else:
		tmp = ((i / n) * math.log(t_0)) / (((i / n) + 1.0) - 1.0)
	return 100.0 * ((math.exp((n * tmp)) - 1.0) / (i / n))

Julia

function code(i, n)
	t_0 = Float64(1.0 + Float64(i / n))
	tmp = 0.0
	if (t_0 == 1.0)
		tmp = Float64(i / n);
	else
		tmp = Float64(Float64(Float64(i / n) * log(t_0)) / Float64(Float64(Float64(i / n) + 1.0) - 1.0));
	end
	return Float64(100.0 * Float64(Float64(exp(Float64(n * tmp)) - 1.0) / Float64(i / n)))
end

MATLAB

function tmp_2 = code(i, n)
	t_0 = 1.0 + (i / n);
	tmp = 0.0;
	if (t_0 == 1.0)
		tmp = i / n;
	else
		tmp = ((i / n) * log(t_0)) / (((i / n) + 1.0) - 1.0);
	end
	tmp_2 = 100.0 * ((exp((n * tmp)) - 1.0) / (i / n));
end

Wolfram

code[i_, n_] := Block[{t$95$0 = N[(1.0 + N[(i / n), $MachinePrecision]), $MachinePrecision]}, N[(100.0 * N[(N[(N[Exp[N[(n * If[Equal[t$95$0, 1.0], N[(i / n), $MachinePrecision], N[(N[(N[(i / n), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i / n), $MachinePrecision] + 1.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(i / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024255 
(FPCore (i n)
  :name "Compound Interest"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((lnbase (if (== (+ 1 (/ i n)) 1) (/ i n) (/ (* (/ i n) (log (+ 1 (/ i n)))) (- (+ (/ i n) 1) 1))))) (* 100 (/ (- (exp (* n lnbase)) 1) (/ i n)))))

  (* 100.0 (/ (- (pow (+ 1.0 (/ i n)) n) 1.0) (/ i n))))